A path from s to t on a polyhedral terrain is descending if the height of a point p never increases while we move p along the path from s to t. We introduce a generalization of the shortest descending path problem, called the shortest gently descending path (SGDP) problem, where a path descends, but not too steeply. The additional constraint to disallow a very steep descent makes the paths more...

p-Adic generalization of the Feynman path integrals in quantum mechanics is considered. The probability amplitude Kv(x, t′′; x′, t′) (v = ∞, 2, 3, · · · , p, · · ·) for a particle in a constant field is calculated. Path integrals over Qp have the same form as those over R.

Given a weighted graph G = (V ,E) with a source s and a destination t , a traveler has to go from s to t . However, some of the edges may be blocked at certain times, and the traveler only observes that upon reaching an adjacent site of the blocked edge. Let P = {PG(s, t)} be the set of all paths from s to t . The risk of a path is defined as the longest travel under the assumption that any edg...

Let S be a set of n points in IR and let t > 1 be a real number. A t-spanner for S is a directed graph having the points of S as its vertices, such that for any pair p and q of points there is a path from p to q of length at most t times the Euclidean distance between p and q. Such a path is called a t-spanner path. The spanner diameter of such a spanner is defined as the smallest integer D suc...

Let G = (V,E) be a simple, undirected graph. Given an integer r ≥ 1, we say that G is r-twin-free (or r-identifiable) if the balls B(v, r) for v ∈ V are all different, where B(v, r) denotes the set of all vertices which can be linked to v by a path with at most r edges. These graphs are precisely the ones which admit r-identifying codes. We show that if a graph G is r-twin-free, then it contain...

Let S be a set of n points in IRd and let t > 1 be a real number. A t-spanner for S is a directed graph having the points of S as its vertices, such that for any pair p and q of points there is a path from p to q of length at most t times the Euclidean distance between p and q. Such a path is called a t-spanner path. The spanner diameter of such a spanner is de ned as the smallest integer D suc...

A path to high assurance solutions C r y p t o l : T h e L a n g u a g e o f C r y p t o g r a p h y

A continuous map between topological f : X → Y is said to satisfy the path-lifting property if for any path p : [0, 1] → Y and any x ∈ f(p(0)) there exists a lifting p̃ of the path p with the intitial value x, i.e. there exists a path p̃ such that f ◦ p̃ = p and p̃(0) = x. Similarly, a smooth map between Riemannian manifolds f : X → Y is said to satisfy the rectifiable path-lifting property if the ...